Possible Worlds

Anne is working at her desk. While she is directly aware only of her
immediate situation — her being seated in front of her
computer, the music playing in the background, the sound of her
husband's voice on the phone in the next room, and so on — she
is quite certain that this situation is only part of a series of
increasingly more inclusive, albeit less immediate, situations: the
situation in her house as a whole, the one in her neighborhood, the
city she lives in, the state, the North American continent, the
Earth, the solar system, the galaxy, and so on. On the face of it,
anyway, it seems quite reasonable to believe that this series has a
limit, that is, that there is a maximally inclusive
situation encompassing all others: things, as a whole or,
more succinctly, the actual world.

Most of us also believe that things, as a whole, needn't have been
just as they are. Rather, things might have been different in
countless ways, both trivial and profound. History, from the very
beginning, could have unfolded quite other than it did in fact: The
matter constituting a distant star might never have organized well
enough to give light; species that survived might just as well have
died off; battles won might have been lost; children born might never
have been conceived and children never conceived might otherwise have
been born. In any case, no matter how things had gone they would
still have been part of a single, maximally inclusive,
all-encompassing situation, a single world. Intuitively, then, the
actual world of which Anne's immediate situation is a part is only
one among many possible worlds.

The idea of possible worlds is evocative and appealing. However,
possible worlds failed to gain any real traction among philosophers
until the 1960s when they were invoked to provide the conceptual
underpinnings of some powerful developments in modal logic. Only then
did questions of their nature become a matter of the highest
philosophical importance. Accordingly, Part 1 of this article
will provide an overview of the role of possible worlds in the
development of modal logic. Part 2 explores three prominent
philosophical approaches to the nature of possible
worlds.[1]
Although many of the finer philosophical points of Part 2
do presuppose the technical background of Part 1, the general
philosophical landscape laid out in Part 2 can be appreciated
independently of Part 1.

Although ‘possible world’ has been part of the
philosophical lexicon at least since Leibniz, the notion became
firmly entrenched in contemporary philosophy with the development of
possible world semantics for the languages of modal logic. In
addition to the usual Boolean sentence operators of
classical logic
such as ‘and’ (‘∧’), ‘or’
(‘∨’), ‘not’ (‘¬’), and
‘if...then’ (‘→’), these languages
contain operators intended to represent the modal adverbs
‘necessarily’ (‘□’) and
‘possibly’ (‘◇’). Although a prominent
aspect of logic in both Aristotle's work and the work of many
medieval philosophers, modal logic was largely ignored from the
modern period to the mid-20th century. And even though a variety of
modal deductive systems had in fact been rigorously developed in the
early 20th century, notably by Lewis and Langford (1932), there was
for the languages of those systems nothing comparable to the elegant
semantics that Tarski had provided for the languages of classical
first-order predicate logic. Consequently, there was no rigorous account of
what it means for a sentence in those languages to be true
and, hence, no account of the critical semantic notions of validity
and logical consequence to underwrite the corresponding deductive notions of
theoremhood and provability. A concomitant philosophical casualty of
this void in modal logic was a deep skepticism, voiced most
prominently by Quine, toward any appeal to modal notions in
metaphysics generally, notably, the notion of an essential property.
(See Quine 1953 and 1956, and the appendix to Plantinga 1974.) The
purpose of the following two subsections is to provide a simple and
largely ahistorical overview of how possible world semantics fills
this void; the final subsection presents two important applications
of the semantics. (Readers familiar with basic possible world
semantics can skip to §2 with no significant loss of
continuity.)

Since the middle ages at least, philosophers have recognized a
semantical distinction between extension and
intension. The extension of a denoting expression, or
term, such as a name or a definite description is its
referent, the thing that it refers to; the extension of a predicate
is the set of things it applies to; and the extension of a sentence
is its truth value. By contrast, the intension of an expression is
something rather less definite — its sense, or
meaning, the semantical aspect of the expression that
determines its extension. For purposes here, let us say that a
logic is a formal language together with a semantic theory for
the language, that is, a theory that provides rigorous definitions of
truth, validity, and logical consequence for the
language.[2]
A logic is extensional if the truth value of every sentence
of the logic is determined entirely by its form and the extensions of
its component sentences, predicates, and terms. An extensional logic
will thus typically feature a variety of valid substitutivity
principles. A substitutivity principle says that, if two
expressions are coextensional, that is, if they have the same
extension, then (subject perhaps to some reasonable conditions)
either can be substituted for the other in any sentence salva
veritate, that is, without altering the original sentence's truth
value. In an
intensional logic,
the truth values of some sentences are determined by something over
and above their forms and the extensions of their components and, as
a consequence, at least one classical substitutivity principle is
typically rendered invalid.

Extensionality is a well known and generally cherished feature of
classical propositional and predicate logic. Modal logic, by
contrast, is intensional. To illustrate: the substitutivity principle
for sentences tells us that sentences with the same truth value can
be substituted for one another salva veritate. So suppose that
John's only pets are two dogs, Algol and BASIC, say, and consider two
simple sentences and their formalizations (the predicates in question
indicating the obvious English counterparts):

All John's dogs are mammals: ∀x(Dx →
Mx).

All John's pets are mammals: ∀x(Px →
Mx)

As both sentences are true, they have the same extension. Hence, in
accordance with the classical substitutivity principle for sentences,
we can replace the occurrence of (1) with (2) in the false sentence

Not all John's dogs are mammals: ¬∀x(Dx
→ Mx)

and the result is the equally false sentence

Not all John's pets are mammals: ¬∀x(Px
→ Mx).

However, when we make the same substitution in the true sentence

Necessarily, all John's dogs are mammals: □∀x(Dx → Mx),

the result is the sentence

Necessarily, all John's pets are mammals: □∀x(Px → Mx),

which is intuitively false, as John surely could have had a
non-mammalian pet. In a modal logic that accurately represents the
logic of the necessity operator, therefore, the substitutivity
principle for sentences will have to fail.

The same example illustrates that the substitutivity principle for
predicates will have to fail in modal logic as well. For, according
to our example, the predicates ‘D’ and
‘P’ that are true of John's dogs and of John's
pets, respectively, are coextensional, i.e.,
∀x(Dx ↔ Px). However, while
substituting the latter predicate for the former in (3) results in a
sentence with the same truth value, the same substitution in (5) does
not.

Modal logic, therefore, is intensional: in general, the truth
value of a sentence is determined by something over and above its
form and the extensions of its components. Absent a rigorous semantic
theory to identify the source of its intensionality and to
systematize intuitions about modal truth, validity, and logical
consequence, there was little hope for the widespread acceptance of
modal logic.

The idea of possible worlds raised the prospect of extensional
respectability for modal logic, not by rendering modal logic itself
extensional, but by endowing it with an extensional semantic theory
— one whose own logical foundation is that of classical
predicate logic and, hence, one on which possibility and necessity
can ultimately be understood along classical Tarskian lines.
Specifically, in possible world semantics, the modal operators
are interpreted as quantifiers over possible worlds, as
expressed informally in the following two general principles:

Nec

A sentence of the form ⌈Necessarily,
φ⌉ (⌈◻φ⌉) is true if and
only if φ is true in every possible
world.[3]

Poss

A sentence of the form ⌈Possibly, φ⌉
(⌈◇φ⌉) is true if and only if φ is true
in some possible world.

Given this, the failures of the classical substitutivity principles
can be traced to the fact that modal operators, so interpreted,
introduce contexts that require subtler notions of meaning for
sentences and their component parts than are
provided in classical logic; in particular, a subtler notion (to be
clarified shortly) is required for predicates than that of the set of
things they happen to apply to.

Tarskian Semantics. Standard model theoretic semantics for the
languages of predicate logic deriving from the work of Tarski (1933,
1944) is the paradigmatic semantic theory for extensional logics.
Given a standard first-order language ℒ and a set D for
the quantifiers of ℒ to range over (typically, some set of
things that ℒ has been designed to describe), a Tarskian
interpretationMfor ℒ assigns, to each
term (constant or variable) τ of ℒ, a referent
aτ ∈ D and, to each n-place
predicate π of ℒ, an appropriate extension
Eπ — a truth value (TRUE or FALSE) if
n = 0, a subset of D if n = 1, and a set of
n-tuples of members of D if n > 1. Given these
assignments, sentences are evaluated as true under the interpretation M — trueM, for short — according to a more or less familiar set of clauses. To facilitate the definition, let M[ν/a] be the interpretation that assigns the individual
a to the variable ν and is otherwise exactly like M.
Then we have:

An atomic sentence
⌈πτ1...τn⌉
(of ℒ) is trueM if and only if

n = 0 (i.e., π is a sentence letter) and the
extension of π is the truth value TRUE; or

n = 1 and aτ1 is
in the extension of π; or

n > 1 and 〈aτ1,
..., aτn〉 is
in the extension of π.

A negation ⌈¬ψ⌉ is trueM if and only if ψ is not trueM.

A material conditional⌈ψ → θ⌉ is trueM
iff, if ψ is trueM, then θ is trueM.

A universally quantified sentence ⌈∀νψ⌉ is
trueM if and only if, for all individuals a ∈ D, ψ is
trueM[ν/a].[4]

Clauses for the other standard Boolean operators and the existential
quantifier under their usual definitions follow straightaway from
these clauses. In particular, where

∃νφ =def ¬∀ν¬φ

it follows that:

An existentially quantified sentence
⌈∃νψ⌉ is is
trueM if and only if, for some individual a ∈ D, ψ is
trueM[ν/a].

It is easy to verify that, in each of the above cases, replacing one
coextensional term, predicate, or sentence for another has no effect
on the truth values rendered by the above clauses, thus guaranteeing
the validity of the classical substitutivity principles and, hence,
the extensionality of first-order logic with a Tarskian semantics.

From Tarskian to Possible World Semantics. The truth
conditional clauses for the three logical operators directly reflect
the meanings of the natural language expressions they symbolize:
‘¬’ means not; ‘→’ means
if...then; ‘∀’ means all. It is
easy to see, however, that we cannot expect to add an equally simple
clause for sentences containing an operator that symbolizes
necessity. For a Tarskian interpretation fixes the domain of
quantification and the extensions of all the predicates. Pretty
clearly, however, to capture necessity and possibility, one must be
able to consider alternative “possible” domains of
quantification and alternative “possible” extensions for
predicates as well. For, intuitively, under different circumstances,
fewer, more, or other things might have existed and things that
actually exist might, in those circumstances, have had very different
properties. (6), for example, is false because John could have had
non-mammalian pets: a canary, say, or a turtle, or, under
very different circumstances, a dragon. A bit more formally
put: Both the domain of quantification and the extension of the
predicate ‘P’ could, in some sense or other, have
been different.

Possible world semantics, of course, uses the concept of a possible
world to give substance to the idea of alternative extensions and
alternative domains of quantification. (Possible world semantics can
be traced most clearly back to the work of Carnap (1947), its basic
development culminating in the work of Hintikka (1957, 1961), Bayart
(1958, 1959), and Kripke (1959, 1963a, 1963b).[5]) As in Tarskian semantics, a
possible world interpretation M of a modal language ℒ
starts with nonempty set D, although thought of now as the set
of “possible individuals” of M. Also as in Tarskian
semantics, M assigns each term τ of ℒ a referent
aτ in D.[6] Additionally however, M contains a
set W, the set of “possible worlds” of M,
one of which is designated its “actual world”, and each
world w in W is assigned its own domain of
quantification, d(w) ⊆ D, intuitively, the
set of individuals that exist in w.[7] To capture the idea of both the actual and
possible extensions of a predicate, I assigns to each
n-place predicate π a function Iπ
— the intension of π — that, for each possible
world w, returns the extension
Iπ(w) of π atw: a
truth value, if n = 0; a set of individuals, if n = 1;
and a set of n-tuples of individuals, if n > 1.[8] We can thus rigorously
define a “possible extension” of a predicate π to be
any of its w-extensionsIπ(w), for any world w.

The Tarskian truth conditions above are now generalized by
relativizing them to worlds as follows: for any possible world
w (the world of evaluation):

An atomic sentence
⌈πτ1...τn⌉
(of ℒ) is trueMatw if and only if:

n = 0 and the w-extension of π is the truth
value TRUE; or

n = 1 and aτ1 is in
the w-extension of π; or

n > 1 and
〈aτ1,...,
aτn〉 is in the
w-extension of π.

A negation ⌈¬ψ⌉ is
trueM at w if and only ψ is not
trueM in w.

A material conditional⌈ψ→θ⌉ is trueM
at w iff, if ψ is trueM at w, then θ is trueM at w.

A quantified sentence ⌈∀νψ⌉ is trueM at
w if and only if, for all individuals a that
exist in w, ψ is trueM[ν/a].

And to these, of course, is added the critical modal case that
explicitly interprets the modal operator to be a quantifier over
worlds, as we'd initially anticipated informally in our principle
Nec:

A necessitation
⌈◻ψ⌉ is
trueM at w if and only if, for all
possible worlds u of M, ψ is
trueM
at u.[9]

A sentence φ is falseM at w
just in case it is not trueM at w, and
φ is said to be trueM just in case
φ is trueM at the actual world of
M.

On the assumption that there is a (nonempty) set of all possible
worlds and a set of all possible individuals, we can define
“objective” notions of truth at a world and of truth
simpliciter, that is, notions that are not simply
relative to formal, mathematical interpretations but, rather,
correspond to objective reality in all its modal glory. Let
ℒ be a modal language whose names and predicates represent
those in some fragment of ordinary language (as in our examples
(5) and (6) above). Say that M is the
“intended” interpretation of ℒ if (i) its set
W of “possible worlds” is in fact the set of
all possible worlds, (ii) its designated “actual
world” is in fact the actual world, (iii) its set D
of “possible individuals” is in fact the set of all
possible individuals, and (iv) the referents assigned to the
names of ℒ and the intensions assigned to the predicates
of ℒ are the ones they in fact have. Then, where M
is the intended interpretation of ℒ, we can say that
a sentence φ of ℒ is
true at a possible world w just in case φ
is trueM at w, and
that φ is true just in case it is
trueM at the actual world. (Falsity at
w and falsity, simpliciter, are defined accordingly.) Under the
assumption in question, then, the modal clause above takes on pretty much the
exact form of our informal principle Nec.

Call the above basic possible world semantics. Spelling
out the truth conditions for (6) (relative to the
intended interpretation of its language), basic
possible world semantics tells us that (6) is
true if and only if

For all possible worlds w, ‘∀x(Px → Mx)’ is true at w.

And by unpacking (8) in terms of the quantificational, material conditional,
and atomic clauses above we have that (6) is true if and only if

For all possible worlds w, and for all possible individuals
a that exist in w, if a is in the
w-extension of ‘P’
then a is in the w-extension of ‘M’.

Since we are evaluating (6) with regard to the intended interpretation of its
language, the w-extension of ‘P’ that is returned by its intension, for any world w, is the (perhaps empty) set of John's pets
in w and that of ‘M’ is the set of mammals in w. Hence, if w is a world where John has a pet
canary — COBOL, say — COBOL is in the w-extension
of ‘P’ but not that of ‘M’ ,
i.e., ‘∀x(Px → Mx)’ is
false at w and, hence, by the truth condition (9), (6) is false at the actual world — that is, (6) is false simpliciter, as it should be.

Note that interpreting modal operators as quantifiers over possible
worlds provides a nice theoretical justification for the usual
definition of the possibility operator in terms of necessity,
specifically:

⌈◇φ⌉
=def⌈¬◻¬φ⌉.

That is, a sentence is possible just in case its negation isn't
necessary. Since, semantically speaking, the necessity
operator is literally a universal quantifier, the definition
corresponds exactly to the definition
(7)
of the existential quantifier. For, unpacking the right side of
definition (10) according to the negation and necessitation clauses
above (and invoking the definitions of truth and truth at a world
simpliciter), we have:

⌈◇φ⌉ is true iff it is not
the case that, for all possible worlds w, φ is not true at w.

Clearly, however, if it is not the case that φ fails to be true
at all possible worlds, then it must be true at some world; hence:

for which (12) and the possible world truth conditions for
quantified, Boolean, and atomic sentences yield the correct truth condition:

There is a possible world w and an individual
a existing in w that is in the
w-extension of ‘P’ but not
that of ‘M’,

that is, less stuffily, there is a possible world in which, among
John's pets, at least one is not a mammal.

Summary: Intensionality and Possible Worlds. Analyzed in terms
of possible world semantics, then, the general failure of classical
substitutivity principles in modal logic is due, not to an
irreducibly intensional element in the meanings of the modal
operators, but rather to a sort of mismatch between the surface
syntax of those operators and their semantics: syntactically, they
are unary sentence operators like negation; but semantically, they
are, quite literally, quantifiers. Their syntactic similarity to
negation suggests that, like negation, the truth values of
⌈□φ⌉ and
⌈◇φ⌉, insofar as they are
determinable at all, must be determined by the truth value of φ.
That they are not (in general) so determined leads to the distinctive
substitutivity failures noted above. The possible worlds analysis of
the modal operators as quantifiers over worlds reveals that the unary
syntactic form of the modal operators obscures a semantically
relevant parameter. When the modal operators are interpreted as
quantifiers, this parameter becomes explicit and the reason
underlying the failure of extensionality in modal logic becomes
clear: That the truth values of
⌈□φ⌉ and
⌈◇φ⌉ are not in general
determined by the truth value of φ at the world of evaluation is,
semantically speaking, nothing more than the fact that the truth
values of ‘∀xFx’ and
‘∃xFx’ are not in general determined by the
truth value of ‘Fx’, for any particular value of
‘x’. Possible world semantics, therefore,
explains the intensionality of modal logic by revealing that
the syntax of the modal operators prevents an adequate expression of
the meanings of the sentences in which they occur. Spelled out as
possible world truth conditions, those meanings can be expressed in a
wholly extensional fashion.
(For
a more formal exposition of this point, see the supplemental article
The Extensionality of Possible World Semantics.)

As noted, the focus of the present article is on the metaphysics of
possible worlds rather than applications. Of course, the semantics of
modal languages is itself an application, but one that is of singular
importance, both for historical reasons and because most applications
are in fact themselves applications of (often extended or modified
versions of) the semantical apparatus. Two particularly important
examples are the analysis of intensions and a concomitant explication
of the de re/de dicto
distinction.[10]

The Analysis of Intensions. As much a barrier to the
acceptance of modal logic as intensionality itself was the need to
appeal to intensions per se — properties, relations,
propositions, and the like — in semantical explanations.
Intensional entities have of course featured prominently in the
history of philosophy since Plato and, in particular, have played
natural explanatory roles in the analysis of
intentional attitudes
like
belief
and
mental content.
For all their prominence and importance, however, the nature of these
entities has often been obscure and controversial and, indeed, as a
consequence, they were easily dismissed as ill-understood and
metaphysically suspect “creatures of darkness” (Quine
1956, 180) by the naturalistically oriented philosophers of the
early- to mid-20th century. It is a virtue of possible world
semantics that it yields rigorous definitions for
intensional entities. More specifically, as described above, possible
world semantics assigns to each n-place predicate π a
certain function Iπ — π's intension
— that, for each possible world w, returns the extension
Iπ(w) of π at w. We can define
an intension per se, independent of any language, to be any
such function on worlds. More specifically:

A proposition is any function from worlds to truth
values.

A property is any function from worlds to sets of
individuals.

An n-place relation (n > 1) is any
function from worlds to sets of n-tuples of
individuals.

The adequacy of this analysis is a matter of lively debate that
focuses chiefly upon whether or not intensions, so defined, are too
“coarse-grained” to serve their intended purposes. (See,
e.g., Stalnaker 1987 and 2012 for a strong defense of the analysis.)
However, Lewis (1986, §1.5) argues that, even if the above
analysis fails for certain purposes, it does not follow that
intensions cannot be analyzed in terms of possible worlds, but only
that more subtle constructions might be required. This reply appears
to side-step the objections from granularity while preserving the
great advantage of the possible worlds analysis of intensions, viz.,
the rigorous definability of these philosophically significant
notions.

The De Re / De Dicto Distinction. A particularly
rich application of the possible world analysis of intensions
concerns the analysis of the venerable distinction between de
re and de dicto
modality.[11]
Among the strongest modal intuitions is that the possession of a
property has a modal character — that things exemplify, or fail
to exemplify, some properties necessarily, or
essentially, and others only accidentally. Thus, for
example, intuitively, John's dog Algol is a pet accidentally; under
less fortunate circumstances, she might have been, say, a stray that
no one ever adopted. But she is a dog essentially; she couldn't have
been a flower, a musical performance, a crocodile or any other kind of
thing.

Spelling out this understanding in terms of worlds and the preceding
analysis of intensions, we can say that an individual a has a
property F essentially if a has F in every world
in which it exists, that is, if, for all worlds w in which
a exists, a ∈ F(w). Likewise,
a has F accidentally if a has F in the
actual world @ but lacks it in some other world, that is, if a
∈ F(@) but, for some world w in which a
exists, a ∉ F(w). Thus, let
‘G’ and ‘T’ symbolize ‘is
a dog’ and ‘is someone's pet’, respectively; then,
where ‘E!x’ is short for
‘∃y(x=y)’ (and, hence,
expresses that x exists), we have:

Algol is a dog essentially: □(E!a →
Ga)

Algol is a pet accidentally: Ta ∧
◇(E!a ∧ ¬Ta)

More generally, sentences like (15) and (16) in which properties are
ascribed to a specific individual in a modal context — signaled
formally by the occurrence of a name or the free occurrence of a
variable in the scope of a modal operator — are said to exhibit
modality de
re[12]
(modality of the thing). Modal sentences that do not, like

Necessarily, all dogs are mammals: □∀x(Gx → Mx)

are said to exhibit modality de dicto (roughly, modality
of theproposition).
Possible world semantics provides an illuminating analysis of the key
difference between the two: The truth conditions for both modalities
involve a commitment to possible worlds; however, the truth
conditions for sentences exhibiting modality de re involve in
addition a commitment to the meaningfulness of
transworld identity,
the thesis that, necessarily, every individual (typically, at any
rate) exists and exemplifies (often very different) properties in
many different possible worlds. More specifically, basic possible
world semantics yields intuitively correct truth values for sentences
of the latter sort by (i) permitting world domains to overlap and
(ii) assigning intensions to predicates, thereby, in effect,
relativizing predicate extensions to worlds. In this way, one and the
same individual can be in the extension of a given predicate at all
worlds in which they exist, at some such worlds only, or at none at
all. (For further discussion, see the entry on
essential vs. accidental properties.)

The power and appeal of basic possible world semantics is undeniable.
In addition to providing a clear, extensional formal semantics for a
formerly somewhat opaque, intensional notion, cashing possibility as
truth in some possible world and necessity as truth in every such
world seems to tap into very deep intuitions about the nature of
modality and the meaning of our modal discourse. Unfortunately, the
semantics leaves the most interesting — and difficult —
philosophical questions largely unanswered. Two arise with particular
force:

QW

What, exactly, is a possible world?

And, given QW:

QE

What is it for something to exist in a possible world?

In this section we will concern ourselves with, broadly speaking, the
three most prominent philosophical approaches to these
questions.[13]

Recall the informal picture that we began with: a world is, so to
say, the “limit” of a series of increasingly more
inclusive situations. Fleshed out philosophical accounts of this
informal idea generally spring from rather different intuitions about
what one takes the “situations” in the informal picture
to be. A particularly powerful intuition is that situations are
simply structured collections of physical objects: the
immediate situation of our initial example above, for instance, consists
of, among other things, the objects in Anne's office
— notably Anne herself, her desk and her computer, with her
seated at the former and typing on the latter — and at least
some of the things in the next room — notably, her husband and
the phone he is talking on. On this view, for one situation s
to include another r is simply for r to be a (perhaps
rather complex and distributed) physical part of s. The actual
world, then, as the limit of a series of increasingly more inclusive
situations in this sense, is simply the entire physical universe: all
the things that are some spatiotemporal distance from the objects in
some arbitrary initial situation, structured as they in fact are; and
other possible worlds are things of exactly the same sort. Call this
the concretist intuition, as possible worlds are understood to
be concrete physical situations of a special sort.

The originator and, by far, the best known proponent of concretism is
David Lewis.
For Lewis and, as noted, concretists generally, the actual world is
the concrete physical universe as it is, stretched out in space-time.
As he rather poetically expresses it (1986, 1):

The world we live in is a very inclusive thing....There is
nothing so far away from us as not to be part of our world.
Anything at any distance is to be included. Likewise the
world is inclusive in time. No long-gone ancient Romans, no
long-gone pterodactyls, no long-gone primordial clouds of
plasma are too far in the past, nor are the dead dark stars
too far in the future, to be part of this same
world....[N]othing is so alien in kind as not to be part of
our world, provided only that it does exist at some distance
and direction from here, or at some time before or after or
simultaneous with now.

The actual world provides us with our most salient example of what a
possible world is. But, for the concretist, other possible worlds are
no different in kind from the actual world (ibid., 2):

There are countless other worlds, other very inclusive
things. Our world consists of us and all our surroundings,
however, remote in time and space; just as it is one big
thing having lesser things as parts, so likewise do other
worlds have lesser other-worldly things as parts.

It is clear that spatiotemporal relations play a critical role in
Lewis's conception. However, it is important to note that Lewis
understands such relations in a very broad and flexible way so as to
allow, in particular, for the possibility of spirits and other
entities that are typically thought of as non-spatial; so long as they
are located in time, Lewis writes, “that is good enough”
(ibid., 73). So with this caveat, let us say that that an
object a is connected if any two of its parts bear
some spatiotemporal relation to each other,[14] and that a is
maximal if none of its parts is spatiotemporally related to
anything that is not also one of its parts. Then we have the following
concretist answers to our questions:

It follows from AW1 (and reasonable assumptions) that distinct
worlds do not overlap, spatiotemporally; that no spatiotemporal part
of one world is part of
another.[16]
Moreover, given
Lewis's counterfactual analysis of causation,
it follows from this that objects in distinct worlds bear no causal
relations to one another; nothing that occurs in one world has any
causal impact on anything that occurs in any other world.

Critically, for Lewis, worlds and their denizens do not differ in the
manner in which they exist. The actual world does not enjoy a
kind of privileged existence that sets it apart from other worlds.
Rather, what makes the actual world actual is simply that it is
our world, the world that we happen to inhabit. Other worlds
and their inhabitants exist just as robustly as we do, and in
precisely the same sense; all worlds and all of their denizens are
equally real.[17] A significant semantic corollary of this
thesis for Lewis is that the word ‘actual’ in the phrase
‘the actual world’ does not indicate any special
property of the actual world that distinguishes it from all
other worlds; likewise, an assertion of the form ‘a is
actual’ does not indicate any special property of the individual
a that distinguishes it from the objects existing in other
worlds. Rather, ‘actual’ is simply an indexical
whose extension is determined by the context of utterance. Thus, the
referent of ‘the actual world’ in a given utterance is
simply the world of the speaker, just as the referent of an utterance
of ‘the present moment’ is the moment of the utterance;
likewise, an utterance of the form ‘a is actual’
indicates only that a shares the same world as the speaker.
The speaker thereby ascribes no special property to a but,
essentially, expresses no more than when she utters ‘a is
here’, understood in the broadest possible sense. By
the same token, when we speak of non-actual possibilia —
Lewis's preferred label for the denizens of possible worlds — we
simply pick out those objects that are not here in the
broadest sense. In the mouth of an other-worldly metaphysician, we
here are all among the non-actual possibilia of which she
speaks in her lectures on de re modality.

Modal Reductionism and Counterparts. Lewis parted ways dramatically with his mentor W. V. O. Quine on modality. Quine (1960, §41) stands in a long line of philosophers dating back at least to David Hume who are skeptical, at best, of the idea that modality is an objective feature of reality and, consequently, who question whether modal assertions in general can be objectively true or false, or even coherent. Lewis, by contrast, wholly embraces the objectivity of modality and the coherence of our modal discourse. What he denies, however, is that modality is a fundamentally irreducible feature of the world. Lewis, that is, is a modal
reductionist. For Lewis, modal notions are not
primitive. Rather, truth conditions for modal sentences can be given
in terms of worlds and their parts; and worlds themselves, Lewis
claims, are defined entirely in non-modal terms. The earliest
presentation of Lewis's theory of modality (Lewis 1968) —
reflecting Quine's
method of regimentation
— offers, rather than a possible world semantics, a scheme for
translating sentences in the language of modal predicate logic
into sentences of ordinary first-order logic in which the modal
operators are replaced by explicit quantifiers over
worlds.[18]
The mature account of Lewis 1986 is much more semantic in orientation: it
avoids any talk of translation and offers instead a (somewhat
informal) account of concretist possible world truth conditions for a
variety of modal assertions. Nonetheless, it is useful to
express the logical forms of these truth conditions explicitly in
terms of worlds, existence in a world (in the sense of AE1, of
course), and the counterpart relation, which will be
discussed shortly:

Wx:

x is a
world

Ixy:

xexists in world y

Cxy:

x is a counterpart of y

For sentences like
(17)
involving only de dicto modalities, Lewis's truth conditions
are similar in form to the truth conditions generated by the modal
clauses of basic possible world semantics; specifically, for (17):

For every world w, every individual x in
w that is a dog is a mammal: ∀w(Ww →
∀x(Ixw → (Gx → Mx))).

As in possible world semantics, the modal operators
‘□’ and ‘◇’ “turn
into” quantifiers over worlds in concretist truth
conditions (1986, 5). Also as in possible world semantics,
a quantifier (in effect) ranging over individuals that occurs
in the scope of a quantifier (in effect) ranging over worlds
— ‘∀x’ and
‘∀w’, respectively, in (18) — is, for
each value w of the bound world variable, restricted to the
objects existing in w. However, unlike possible world
semantics, predicates are not to be thought of as having different
extensions at different worlds. Rather, for Lewis, each
(n-place) predicate has a single extension that can contain
(n-tuples of) objects across many different worlds —
intuitively, all of the objects that have the property (or
n-tuples of objects that stand in the relation) expressed by
the predicate across all possible worlds. Thus, in particular, the
predicate ‘G’ picks out, not just this-worldly dogs
but other-worldly canines as well. Likewise, the pet predicate
‘T’ picks out both actual and other-worldly
pets. Such a move is not feasible in basic possible world semantics,
which is designed for a metaphysics in which one and the same
individual can exemplify a given property in some worlds in which they
exist but not others. Hence, a typical predicate will be true of an
individual with respect to some worlds and false of it with respect to
others. But, for Lewis, as we've seen, distinct possible worlds do not
overlap and, hence, objects are worldbound, thereby eliminating the
need to relativize predicate extensions to worlds.

However, this very feature of Lewis's account —
worldboundedness — might appear to threaten its coherence. For
example, since Algol is in fact a pet, given worldboundedness and the
definition
AE1
of existence in a world w, we have:

There is no world w such that Algol exists in
w and fails to be someone's pet:
¬∃w(Iaw ∧ ¬Ta),

But, according to Lewis's analysis, the modal operators
‘□’ and ‘◇’, semantically, are
quantifiers over worlds. Hence, (19) might appear to be exactly the
concretist truth condition for the denial of (the right conjunct of)
(16), i.e., it might appear that, on Lewis's
analysis, Algol is not a pet accidentally but essentially; likewise,
more generally, any individual and any intuitively accidental property
of that individual.

In fact, Lewis whole-heartedly accepts that things have accidental
properties and, indeed, would accept that
(16)
is robustly true. His explanation involves one of the most
interesting and provocative elements of his theory: the doctrine of
counterparts. Roughly, an object y in a world
w2 is a counterpart of an object x in
w1 if y resembles x and nothing else
in w2 resembles x more than
y.[19]
Each object is thus
its own (not necessarily unique) counterpart in the world it inhabits
but will typically differ in important ways from its other-wordly
counterparts. A typical other-worldly counterpart of Algol, for
example, might resemble her very closely up to some point in her
history — a point, say, after which she continued to live out
her life as a stray instead of being brought home by our kindly
dog-lover John. Hence, sentences making de re assertions about
what Algol might have done or what she could or could
not have been are unpacked, semantically, as sentences about her
counterparts in other possible worlds. Thus, when we analyze
(16) accordingly, we have the entirely unproblematic concretist truth
condition:

Algol is a pet, but there is a world in which exists a
counterpart of hers that is not:Ta ∧
∃w(Ww ∧ ∃x(Ixw ∧
Cxa ∧ ¬Tx)).

Ascriptions of essential properties, as in
(15),
are likewise unpacked in terms of counterparts: to say that Algol is
a dog essentially is to say that

All of Algol's counterparts in any world are dogs:
∀w(Ww →
∀x((Ixw ∧ Cxa) →
Gx)).

The Analysis of Intensions. Lewis's possible world truth
conditions are expressed in classical non-modal logic and, hence, they
are to be interpreted by means of standard Tarskian semantics. Thus,
n-place predicates π are assigned extensions
Eπ — in particular, for 1-place predicates,
sets of individuals — as their semantic values, as described in
the exposition in §1.2 above. However,
given worldboundedness and the fact that predicate extensions are
drawn not simply from the actual world but from all possible worlds,
these extensions are able to serve as intensions in Lewis's
theory. As in basic possible world semantics, intensional entities in
general can be defined in terms of the basic ontology of the theory
independent of the linguistic roles they can play as the intensions of
predicates. And because individuals are worldbound, Lewis is able to
simplify the definitions given in §1.3 by defining intensions as sets
rather than functions:

A proposition is any set of worlds.

A property is any set of individuals.

An n-place relation (n > 1) is any set
of n-tuples of individuals.[20]

Thus, on this analysis, a proposition p is true in a
world w just in case w ∈ p and an individual
a has a property P just in case a ∈
P. (Note that propositions are thus simply properties of worlds
on these definitions.) a has Paccidentally
just in case a ∈ P but b ∉ P
for some other-worldly counterpart of b of a; and
a has Pessentially if b ∈ P
for every counterpart b of a.

In Lewis's theory of modality, then, modal operators are understood
semantically to be quantifiers over concrete worlds, predicates
denote intensions understood as sets of (n-tuples of) parts of
those worlds, and sentences involving de re modalities are
understood in terms of counterparts. To the extent that these notions
are free of modality, Lewis has arguably reduced modal notions to
non-modal.

That Lewis's truth conditions for modal statements are themselves free
of modality and, hence, that his theory counts as a genuine reduction
of modal notions to non-modal is not terribly controversial (albeit
not undisputed — see Lycan 1991, 224–27; Divers and Melia
2002, 22–24). Significantly more controversial, and perhaps far
more critical to the project, is whether or not his account is
complete, that is, whether or not, for all modal statements
φ, (i) if φ is intuitively true, then its Lewisian truth
condition holds (ii) if φ is intuitively false, then its Lewisian
truth condition fails.[21]
The challenge to Lewis, then, is that his
account can be considered successful only if it is complete
in this sense.

The chief question Lewis faces in this regard is whether there are
enough worlds to do the job. The truth condition
(20)
for the intuitively true
(16)
says that there exists a possible world in which a counterpart of
Algol is no one's pet. By virtue of what in Lewis's theory does such
a world exist? The ideal answer for Lewis would be that some
principle in his theory guarantees a plenitude of worlds, a
maximally abundant array of worlds that leaves “no gaps in
logical space; no vacancies where a world might have been, but
isn't” (Lewis 1986, 86). From this it would follow that the
worlds required by the concretist truth condition for any intuitive
modal truth exist. Toward this end, Lewis initially considers the
evocative principle:

Ways

Absolutely every way that a world could be is a way
that some world is.

Since, in particular, a world satisfying (20)
seems quite obviously to be a way a world could be, by Ways
such a world exists. But there is a fatal flaw here: Lewis himself
(1973, 84) identifies ways that a world could be with worlds
themselves. So understood, Ways collapses into the triviality
that every world is identical to some
world.[22]

Lewis finds a replacement for Ways in a principle of
recombination whereby “patching together parts of
different possible worlds yields another possible world” (1986,
87–88). The principle has two aspects. The first is the principle
that “anything can coexist with anything”. For “if
there could be a dragon, and there could be a unicorn,” Lewis
writes, “but there couldn't be a dragon and a unicorn side by
side, that would be ... a failure of plenitude” (ibid.,
88). Given that individuals are worldbound, however, the principle is
expressed more rigorously (and more generally) in terms of
other-worldly duplicates:

R1

For any (finite or infinite) number of objects
a1, a2, ..., there is a
world containing any number of duplicates of each of those
objects in any spatiotemporal arrangement
(size and shape permitting).

The second aspect of the principle expresses “the Humean denial
of necessary connections” (ibid., 87), that is, the idea
that anything can fail to coexist with anything else. For
“if there could be a talking head contiguous to the rest of a
living human body, but there couldn't be a talking head separate from
the rest of a human body, that too would be a failure of
plenitude” (ibid). To express this a bit more
rigorously, say that objects a1,
a2, ..., are independent of objects
b1, b2, ..., if no sum of any
parts of the former are parts or duplicates of any sum of any parts
of the latter and vice versa; then we have:

R2

For any world w any (finite or infinite number of)
objects a1, a2, ..., in
w and any objects b1,
b2, ..., in w that are independent of
a1, a2, ..., there is a
world containing duplicates of a1,
a2, ..., and no duplicates of
b1, b2, ... .

Worlds that satisfy the concretist truth conditions for workaday
possibilities like
(16)
are easily conceived as consisting of duplicates of relevant parts of
the actual world — suitably organized to retain their actual
properties, or not, as needed. Hence, the existence of such worlds
does indeed appear to follow from the existence of the actual world
by recombination. Worlds containing talking donkeys, exotic species
resulting from a wholly different evolutionary history, worlds with
silicon-based life forms, and so on present a bigger challenge to the
view. Nonetheless, it is not entirely implausible to think such
worlds exist given suitable duplication and reorganization of
microphysical
objects.[23]

Whether recombination completely captures our modal intuitions
regarding plenitude is still a matter of some dispute.[24] However, even if it
doesn't, it is less than clear whether this counts against the success
of Lewis's reductionist project. For, as a realist about worlds, Lewis
does not seem to be under any obligation to “derive”
plenitude from more fundamental principles. Hence, there is no
obvious reason why he cannot respond to charges of incompleteness by
saying that it is simply a presupposition of his theory that logical
space has no gaps, that there are always enough worlds to satisfy the
concretist truth condition for any intuitive modal truth.[25] So understood, the
role of recombination for a realist about worlds like Lewis is
something like the role of such axioms as powerset and replacement for
a realist about sets: given some sets, these principles provide us
with a detailed — but always less than complete —
characterization of what further sets there are. Their role,
therefore, is to give us insight into the richness and diversity of
set theoretic space, not a complete mechanism for proving which
particular sets do or do not exist. Likewise recombination
vis-à-vis worlds and logical space.

Lewis's theory is particularly commendable for its striking
originality and ingenuity and for the simple and straightforward
answers AW1 and AE1 that it provides to our two
questions QW and QE above. Furthermore, because worlds
are (plausibly) defined entirely in nonmodal terms, the truth
conditions provided by Lewis's translation scheme themselves appear
to be free of any implicit modality. Hence, unlike many other popular
accounts of possible worlds (notably, the abstractionist accounts
discussed in the following section), Lewis's promises to provide a
genuine analysis of the modal operators.

Perhaps the biggest — if not the most philosophically
sophisticated — challenge to Lewis's theory is “the
incredulous stare”, i.e., less colorfully put, the fact that
its ontology is wildly at variance with common sense. Lewis faces
this objection head on: His theory of worlds, he acknowledges,
“does disagree, to an extreme extent, with firm common
sense opinion about what there is” (1986, 133). However, Lewis
argues that no other theory explains so much so economically. With
worlds in one's philosophical toolkit, one is able to provide elegant
explanations of a wide variety of metaphysical, semantical, and
intentional phenomena. As high as the intuitive cost is, Lewis (135)
concludes, the existence of worlds “ought to be accepted as
true. The theoretical benefits are worth it.”

A rather different set of intuitions about situations is that they
are abstract entities of a certain sort: They are
states or conditions, of varying detail and
complexity, that a concrete world could be in — they are
ways that things, as a whole, could
be.[26]
Thus, returning to our original example, one very simple way things
could be is for our philosopher Anne to be in her office. We can now
imagine, as in our example, further detail being successively added
to that description to yield more complex ways things could be: Anne
working at her desk in her office; music being in the background; her
husband being on the phone in the next room; her neighbor mowing the
lawn next door; and so on. Roughly speaking, then, a possible world
for an abstractionist is the limit of such a
“process” of consistently extending and adding detail to
some initial state of the world; it is a total way things
could be, a consistent state of the world that settles every
possibility; a consistent state to which no further detail could be
added without rendering it inconsistent.

To give the notion of a state, or condition, of the world a little
more metaphysical substance, abstractionists typically appeal to more
traditional ontological categories. Thus, for example, that things
could be in the simple state described above might be spelled out in
one of the following ways:

The propositionthat Anne is in her office and
at her desk is possibly true.

The set of propositions {that Anne is in her
office, that Anne is at her desk} is such that,
possibly, all of its members are true.

The propertybeing such that Anne is in her office and
at her desk is possibly exemplified (by “things as a whole”).

Possible worlds are then defined as special cases of the type of
entity in question that are in some relevant sense total.
Adams (1974), for example, defines possible worlds to be consistent
sets of propositions that are total in the sense of containing, for
every proposition p, either p or its negation; Fine
(1977), fleshing out ideas of Prior, defines a possible world to be a
consistent proposition w that is total in the sense that, for
every proposition p, w entails either p or its
negation. For purposes here, however, we will sketch the fundamentals
of the abstractionist view in terms of states of affairs,
following the basic features of the account developed by Plantinga
(1974, 1976), an account that, in the literature, frequently serves as
a particularly trenchant abstractionist counterpoint to Lewis's
concretism.[27]

States of affairs (SOAs) are abstract, intensional entities typically
signified by sentential gerundives like “Algol's being John's
pet” and “There being more than ten solar planets”.
Importantly, SOAs constitute a primitive ontological category for the
abstractionist; they are not defined in terms of possible worlds in
the manner that propositions are in
§1.3.
Just as some propositions are true and others are not, some SOAs are
actual and others are
not.[28]
Note, then, that to
say an SOA is non-actual is not to say that it does not
actually exist. It is simply to say that it is not, in fact,
a condition, or state, that the concrete world is actually in.
However, because ‘____ is actual’ is often used simply to
mean ‘____ exists’, there is considerable potential for
confusion here. So, henceforth, to express that an SOA is actual we
will usually say that it obtains.

An SOA is said to be possible (necessary,
impossible) insofar as it is possible (necessary, impossible)
that it obtain. One SOA s is said to include another
t if, necessarily, s obtains only if t does;
sprecludest if, necessarily, s obtains
only if t doesn't. So, for example, Algol's being John's
pet includes Algol's being someone's pet and precludes
there being no pets. Thus, on the abstractionist's
understanding of a situation as a state or condition of the physical
world rather than a concrete, structured piece of it, the inclusion
of one situation in another is a purely logical relation, not
a
mereological
one. Finally, say that an SOA s is total if, for every
SOA t, s either includes or precludes t.
(Abstractionists often use ‘maximal’ instead of
‘total’, but we have already introduced this term in the
context of concretism.) Abstractionist possible worlds are now
definable straightaway:

AW2

w is a possible world =defw is an SOA that is both
possible and total.[29]

It is easy to see that this definition covers the more intuitive
characterizations of abstract possible worlds above: they are
consistent — i.e., possible — states of the world that
settle every possibility, consistent states to which no further
detail could be added without rendering them inconsistent. Note also
that, for the abstractionist, as with the concretist, the actual
world is no different in kind from any other possible world; all
possible worlds exist, and in precisely the same sense as the actual
world. The actual world is simply the total possible SOA that, in
fact, obtains. And non-actual worlds are simply those total possible
SOAs that do not.

What of existence in such worlds? As we've seen, on Lewis's account,
to exist in a concrete world w is literally to exist in
w, that is, within the spatiotemporal boundaries of
w. Clearly, because SOAs are abstract, individuals cannot
exist in abstractionist worlds in anything like the same literal,
mereological sense. Accordingly, the abstractionist defines existence
in a world simply to be a special case of the inclusion relation:

AE2

Individual aexists in possible world
w =defw includes a's existing.

Unlike concretism, then, abstractionism does not entail that
individuals are worldbound; there is no inconsistency whatever in the
idea that many distinct worlds can include the existence of one and
the same individual. Indeed, typically, abstractionists are staunchly
committed to transworld identity and hold that most any given
individual exists in many possible worlds and, moreover, that
contingent individuals, at least, can exemplify very different
properties from world to world. The abstractionist, therefore, has no
need to appeal to counterparts to understand de re modalities
and can therefore accept the truth conditions for such modalities
given by basic possible world semantics (spelled out, of course, in
terms of his definitions AW2 and AE2). In particular,
he can take the standard possible world truth condition for, e.g.,
the right conjunct of
(16)
at face value: ‘◇(E!a ∧ ¬Ta)’
is true on the abstractionist's approach if and only if there is is a
world in which Algol herself, rather than some counterpart of hers,
exists but fails to be anyone's pet.

It is important to note that the possible worlds of abstractionism do
not yield a reductive analysis of modality. The reason for this is
clear: abstract possible worlds are defined in irreducibly modal terms
— a possible world is an SOA that (among other things)
possibly obtains; or a set of propositions such that it is
possible that all of its members are true; or a property that
is possibly exemplified; and so on. Hence, unpacked in terms
of the abstractionist's definitions, the possible world truth
conditions for modal propositions are themselves irreducibly modal.
For example, when we unpack Plantinga's definition of a possible world
in the semantic clause for sentences of the form
⌈◻ψ⌉ in order to derive
the truth condition for (17),
‘□∀x(Gx → Mx)’, we
end up with this:

For all SOAs w, if (i) possibly, w obtains
and (ii) for all SOAs s, either (a) necessarily,
w obtains only if s does or (b)
necessarily, w obtains only if s doesn't,
then, ‘∀x(Gx →
Mx)’ is true at w.

If we now unpack the modal operators in (22) using the corresponding
truth conditional clauses of standard possible world semantics, the
result will contain further world quantifiers. And spelling out those
world quantifiers in turn using Plantinga’s definition will
re-introduce those same modal operators yet again.

More generally, and a bit more exactly, put: As noted above, the
logical framework of basic possible world semantics is classical
predicate logic. The logical framework of abstractionism is modal
predicate logic. Hence, if possible world semantics is supplemented
with abstractionist definitions of possible worlds, then the logical
framework of possible world semantics becomes modal predicate logic
as well and, as a consequence, the extensionality of the semantics is
lost once again. (This point is expressed somewhat more formally in
the supplemental document
The Intensionality of Abstractionist Possible World Semantics.)
Since, as noted above, the central motivation for possible world
semantics was to deliver an extensional semantics for modal
languages, any motivation for abstractionism as a semantic theory is
arguably undermined.[30]

However, it is not entirely clear that this observation constitutes an
objection to abstractionism. For the abstractionist can argue that the
goal of his analysis is the converse of the reductionist's goal: The
reductionist wants to understand modality in terms of worlds; the
abstractionist, by contrast, wants to understand worlds in terms of
modality. That is, the abstractionist can argue that we begin with a
primitive notion of modality and, typically upon a certain amount of
philosophical reflection, we subsequently discover an intimate
connection to the notion of a possible world, as revealed in the
principles Nec
and Poss. The analysis that the
abstractionist provides is designed to make this connection explicit,
ideally, in such a way that Nec and Poss fall out as
theorems of his theory (see, e.g., Plantinga 1985 and Menzel and Zalta
2014).

Hand in glove with the irreducible nature of modality is the nature
of intensional entities. Concretists define intensional entities in
terms of worlds, as described in
§2.1.3.
Abstractionists, by contrast, define worlds in terms of intensional
entities. This divergence in their choice of ontological primitives
reflects, not only their differing stances toward modality, but also
an important methodological difference with regard to metaphysical
inquiry. The concretist is far more pragmatic; notions of
property, relation, proposition, and the
like play certain roles in our theorizing and are subject to a
“jumble of conflicting desiderata” (Lewis 1986,
54). Within a given theory, any entities that can play those roles
fruitfully for the purposes at hand are justifiably identified with
those notions — regardless of how well they comport with
pre-theoretic intuitions. Thus, Lewis finds it to be a strength of
his position that he is able to adopt the set theoretic definitions
in §2.1.3. By contrast, at least some abstractionists —
Plantinga (1987) perhaps most notably — believe that we have
intuitive, pre-theoretic knowledge of intensional entities that
precludes their being identified with set theoretic constructions of
any
sort.[31]
(See Stalnaker 1976 for a particularly illuminating discussion of the
contrast between concretism and abstractionism with respect to the
treatment intensional entities.)

As was noted in
§2.1.2,
for the concretist, there is no special property of the actual world
— actuality — that distinguishes it, in any
absolute sense, from all of the others; it is simply the world that
we inhabit. For abstractionists, however, actuality
is a special property that distinguishes exactly one
possible world from all others — the actual world is the only
world that happens to obtain; it is the one and only way
things could be that is the way things as a whole, in fact,
are. However, for most abstractionists, the distinctiveness
of the actual world does not lie simply in its actuality but in its
ontological comprehensiveness: the actual world encompasses all that
there is. In a word: most abstractionists are actualists.

Actualism
is the thesis that everything that there is, everything that has
being in any sense, is actual. In terms of possible worlds:
Everything that exists in any world exists in the actual
world.[32]
Possibilism, by contrast, is the denial of actualism; it is the
thesis that there are mere possibilia, i.e., things that are
not actual, things that exist in other possible worlds but fail to
exist in the actual world. Concretists are obviously not actualists
(on their understanding of ‘actual’, at any
rate).[33]
Indeed, for the concretist, since individuals are worldbound,
everything that exists in any nonactual possible world is distinct
from everything in the actual world. However, although possibilism
and abstractionism are entirely compatible — Zalta (1983), for
example, embraces both positions — abstractionists
tend to be actualists. The reason for this is clear: Basic
possible world semantics appears to be committed to possibilism and
abstractionism promises a way of avoiding that commitment.

The specter of possibilism first arises with regard to
non-actual possible worlds, which would seem by definition to
be prime examples of mere possibilia. However, we have just
seen that the abstractionist can avoid this apparent commitment to
possibilism by defining possible worlds to be SOAs of a certain sort.
So defined, non-actual worlds, i.e., worlds that fail to obtain, can
still actually exist. Hence, the commitment of basic possible world
semantics to non-actual worlds does not in itself threaten the
actualist's ontological scruples.

However, the specter of possibilism is not so easily exorcised. For
non-actual worlds are not the only, or even the most compelling,
examples of mere possibilia that seem to emerge out of basic
possible world semantics. For instance, it is quite reasonable to
think that evolution could have taken a very different course (or, if
you like, that God could have made very different creative choices)
and that there could have been individuals — call them
Exotics — that are biologically very different from
all actually existing individuals; so different, in fact, that no
actually existing thing could possibly have been an Exotic. According
to basic possible world semantics, the sentence ‘There could
have been Exotics’ or, more formally,

◇∃xEx

is true just in case there is a world in which
‘∃xEx’ is true, i.e., when all is said and
done, just in case:

There is a possible world w and an individual
a in w such that a is an Exotic in
w,

which, a bit less formally, is simply to say that

Some individual is an Exotic in some possible world.

However, since no actually existing thing could have been an Exotic,
anything that is an Exotic in some possible world cannot be among the
things that exist in the actual world. Thus, the truth conditions
that basic possible world semantics assigns to some of our intuitive
modal beliefs appear to entail that there are non-actual individuals
as well as non-actual possible worlds. Defining possible worlds as
SOAs provided a way for the actualist to embrace non-actual worlds
without compromising her actualism. But how is the actualist to
understand the apparent commitment to non-actual individuals
in such truth conditions as (25)?

Answers that have been given to this question represent a rather deep
divide between actualist abstractionists. On the one hand,
“trace” actualists introduce actually existing entities
into their ontologies that can play the role of mere
possibilia in (25) and its like. Trace actualists come in two
varieties: new actualists and haecceitists. New
actualists like Linsky and Zalta (1996) and Williamson (1998, 2000,
2013) argue that, in fact, all individuals are actually existing,
necessary beings but not all of them are necessarily
concrete. Some concrete individuals — those
traditionally (mis-)categorized as
contingent beings — are only contingently concrete. Likewise,
some non-concrete individuals — those, like possible Exotics,
traditionally (mis-)categorized as
contingently non-actual mere possibilia — are only
contingently
non-concrete.[34]

This novel take on modal metaphysics allows the new actualist to
reinterpret possible world semantics so as to avoid possibilism.
Notably, the domain d(w) of a world w is
understood not as the set of things that exist in w —
for all individuals exist in all worlds — but the set of things
that are concrete in
w.[35]
Hence, for the new actualist, the correct truth condition for
(23)
is:

There is a possible world w and an individual a
that is (i) concrete in w and (ii) an Exotic in
w.

On the other hand, haecceitists like Plantinga introduce special
properties — haecceities — to similar ends. The haecceity
of an individual a is the property of being that very
individual, the property being a. A property is a
haecceity, then, just in case it is possible that it is the
haecceity of some
individual.[36]
It is a necessary truth that everything has a haecceity. More
importantly, for haecceitists, haecceities are necessary beings.
Thus, not only is it the case that, had any particular individual
a failed to exist, its haecceity ha would
still have existed, it is also the case that, for any “merely
possible” individual a, there is an actually existing
haecceity that would have been a's haecceity had a
existed. More generally (and more carefully) put: Necessarily, for
any individual a, (i) a has a haecceity h and
(ii) necessarily, h exists.

Like the new actualists, then, the haecceitist's metaphysics enables
him to systematically reinterpret possible world semantics in such a
way that the truth conditions of modal discourse are expressed solely
in term of actually existing entities of some sort rather than actual
and non-actual individuals. More specifically, for the haecceitist,
the domain d(w) of a world w is taken to be the
set of haecceities that are exemplified in w, that
is, the set of haecceities h such that w includes
h's being exemplified. Likewise, the w-extension
of a (1-place) predicate π is taken to be a set of
haecceities — intuitively, those haecceities that are
coexemplified in w with the property expressed by π. So
reinterpreted, the truth condition for
(23)
is:

There is a possible world w and a haecceity h
that is (i) exemplified in w and (ii) coexemplified
with the property being an Exotic in w.

By contrast, “no-trace”, or strict, actualists
like Prior (1957), Adams (1981), and Fitch (1996) hew closely to the
intuition that, had a contingent individual a failed to exist,
there would have been absolutely no trace, no metaphysical vestige,
of a — neither a itself in some non-concrete
state nor any abstract proxy for a. Hence, unlike trace
actualism, there are no such vestiges in the actual world of objects
that are not actual but only could have been.

The logical consequences for no-trace actualists, however, appear to
be severe; at the least they cannot provide a standard
compositional
semantics for modal languages, according to which (roughly) the
meaning of a sentence is determined by its logical form and the
meanings of its semantically significant constituents. In particular,
if there is nothing to play the role of a “possible
Exotic”, nothing that is, or represents, an Exotic in some
other possible world — a mere possibile, a contingently
non-concrete individual, an unexemplified haecceity — then the
strict actualist cannot provide standard, compositional truth
conditions for quantified propositions like
(23)
that yield the intuitively correct truth value. For, understood
compositionally,
(23)
is true if and only if ‘∃xEx’ is true at
some world w. And that, in turn, is true at w if and
only if ‘Ex’ is true at w for some value of
‘x’. But, as just noted, for the strict actualist,
there is no such value of ‘x’. Hence, for the
strict actualist, ‘Ex’ is false at w for
all values of ‘x’ and, hence,
(23)
is false as well. (These issues are explored in much greater detail
in
§4
of the entry
Actualism.)

Like concretism, abstractionism provides a reasonably clear and
intuitive account of what worlds are and what it is to exist in them,
albeit from a decidedly different perspective. Although, as noted in
§2.2.2,
the fact that modality is a primitive in abstractionist definitions
of possible worlds arguably compromises its ability to provide semantically
illuminating truth conditions for the modal operators, those definitions
can be taken to illuminate the connection between our basic modality concepts
and the evocative notion of a possible world that serves as such a
powerful conceptual tool for
constructing philosophical arguments and for analyzing and developing
solutions to philosophical problems. In this regard, particularly
noteworthy are: Plantinga's (1974) influential work on
the ontological argument
and the free will defense against
the problem of evil;
Adams' (1974, 1981) work on
actualism
and actuality; and Stalnaker's (1968, 1987) work on
counterfactual conditionals
and
mental content.

A number of important objections have been voiced in regard to
abstractionism. Some of these are addressed in the document
Problems with Abstractionism.

As its name might suggest, our third approach —
combinatorialism — takes possible worlds to be
recombinations, or rearrangements, of certain metaphysical
simples. Both the nature of simples and the nature of recombination
vary from theory to theory. Quine (1968) and Cresswell (1972), for
example, suggest taking simples to be space-time points (modeled,
perhaps, as triples of real numbers) and worlds themselves to be
arbitrary sets of such points, each set thought of intuitively as a
way that matter could be distributed throughout space-time. (A world
w, so construed, then, is actual just in case a
space-time point p is a member of w if and only if
p is occupied by matter.) Alternatively, some philosophers
define states a world could be in, and possible worlds themselves,
simply to be maximally consistent sets of sentences[37] in an expressively
rich language — “recombinations”, certainly, of the
sentences of the language. (Lewis refers to this view as
linguistic ersatzism.[38]) However, the predominant version of
combinatorialism finds its origins in Russell's (1918/1919) theory of
logical atomism and Wittgenstein's
(1921, 1922, 1974) short but enormously influential Tractatus Logico-Philosophicus.
A suggestive paper by Skyrms (1981) spelling out some of the ideas in
the Tractatus, in turn, inspired a rich and sophisticated
account that is developed and defended in great detail in an important
series of books and articles by D. M. Armstrong (1978a, 1978b, 1986a,
1989, 1997, 2004b, 2004c). In this section, we present a somewhat
simplified version of combinatorialism that draws primarily upon
Armstrong's work. Unless otherwise noted, this is what we shall mean
by ‘combinatorialism’ for the remainder.

Wittgenstein famously asserted that the world is the totality of
facts, not of things (ibid., §1.1). The
combinatorialist spells out Wittgenstein's aphorism explicitly in
terms of an ontology of objects (a.k.a., particulars), universals
(a.k.a., properties and relations), and facts. Facts are either
atomic or molecular. Every atomic fact — Sachverhalt, in
the language of the Tractatus — is
“constituted” by an n-place relation (= property,
for n=1) and n objects that stand in, or
exemplify, that relation. Thus, for example, suppose that
John is 1.8 meters tall. Then, in addition to John and the property
being 1.8 meters tall, there is for the combinatorialist the
atomic fact of John's exemplifying that property. More generally,
atomic facts exist according to the following principle:

Say that the ai are the constituent
objects of the fact in question and R its constituent
universal, and that R and the ai
all exist in
[R,a1,...,an].

A fact is monadic if its constituent universal is a
property. A molecular fact f is a conjunction of
atomic facts. Its constituent objects and universals are exactly
those of its conjuncts and an entity exists in f just in case
it exists in one of its conjuncts. (For simplicity, we stipulate that
an atomic fact has (only) itself as a conjunct and, hence, is
“trivially” molecular.) One fact fincludes
another g if every conjunct of g is a conjunct of
f. (Note, importantly, that inclusion, so defined, is quite
different from the homonymous notion defined in the discussion of
abstractionism above — most notably, combinatorial inclusion is
not a modal notion.) For purposes below, say that an object
a is a bare particular in a molecular fact f
if there is no monadic conjunct of f of which a is the
constituent object, no conjunct of the form a exemplifies F,
for some property F. a is a bare particular if
it is bare in every molecular fact. Intuitively, of course, a bare
particular is an unpropertied object.

There is no upper bound on the “size” of a molecular fact
and no restriction on which atomic facts can form a conjunction; for
any atomic facts at all, there is a molecular fact whose conjuncts
are exactly those facts. As a first cut, then, we can spell out
Wittgenstein's characterization of the (actual) world as the totality
of facts by defining the world to be the largest molecular fact, the
molecular fact that includes all of the atomic
facts.[39]

Although objects and universals are typically included along with
facts in the basic ontology of combinatorialism, facts are typically
considered more fundamental. Indeed, taking his queue from the
Tractarian thesis that the world consists of facts, not things,
Armstrong (1986a, 577) argues that facts alone are ontologically
basic and that objects and universals are simply “aspects of,
abstractions from” facts. Accordingly, he calls the object
constituent of a fact of the form [P,a] a
“thin” particular, an object “considered in
abstraction from all its [intrinsic] properties” (1993, 433);
and where N is conjunction of “all the non-relational
properties of that particular (which would presumably include
P), the atomic fact a's exemplifying N itself is the
corresponding “thick” particular ” (ibid.,
434 — we will occasionally use italics to distinguish a thin
particular a from the corresponding thick particular
a). Though not all combinatorialists of every stripe buy into
Armstrong's “factualist” metaphysics (Bricker 2006), they
do generally agree that facts are more fundamental, at least to the
extent that both the notion of a bare particular, i.e., an object
exemplifying no properties, and that of an unexemplified property are
considered incoherent; insofar as they exist at all, the existence of
both particulars and universals depends on their
“occurring” in some fact or other. Whatever their exact
ontological status, it is an important combinatorialist thesis that
exactly what objects and universals exist is ultimately a
matter for natural science, not metaphysics, to decide.

Objects can be either simple or complex. An object is simple
if it has no proper parts, and complex otherwise. Like
objects, universals too divide into simple and complex. A universal
is simple if it has no other universal as a constituent, and complex
otherwise. Complex universals accordingly come in two varieties:
conjunctive — the constituents of which are simply its
conjuncts — and structural. A structural universal U is
one that is exemplified by a complex object O, and its
constituents are universals (distinct from U) exemplified by
simple parts of O that are relevant to O's being an
instance of
U.[40]
It is important to note that, for Armstrong, the constituency
relation is not the mereological parthood relation. Rather, complex
universals (hence also complex facts of which they are constituents)
enjoy a “non-mereological mode of composition” (1997,
119–123) that, in particular, allows for a richer conception of
their
structure.[41]
(An assumption of our simplified account here will be that both the
proper part of relation and the constituency relation are
well-founded. It follows that (i) there is no
gunk,
i.e., that every complex object is composed, ultimately, entirely of
simples and (ii) complex universals — hence the complex facts
in which they are exemplified — are ultimately
“grounded” in simple facts, i.e., that they cannot be
infinitely decomposed into further complex
universals/facts.[42])

To illustrate the basic idea: in Figure 1, the left-hand diagram
depicts a water molecule W comprising an oxygen atom o
and two hydrogen atoms h1 and h2.
For the combinatorialist, “thick” particulars like the
molecule itself as well as its constituent atoms are themselves
facts: o is the fact [O,o] in which the
universal oxygen (O) is exemplified by a thin
particular
o;[43]
likewise h1 and h2. W in
turn comprises those monadic facts and the relational facts
[B,o,h1],
[B,o,h2] wherein the covalent bonding
relation B holds between the oxygen atom and the two hydrogen
atoms. The structural universal Water itself, then, shares
this structure — it is, so to say, an isomorph
consisting of the monadic universals O and H and the
binary relation B, structured as indicated in the right-hand
diagram of Figure
1.[44]

It should be clear from
principle AF
that all atomic facts hold; that is, all of them reflect
actual exemplification relations. Obviously, however, possibility
encompasses more than what is actual, that is, there are
possible facts as well as actual facts; the world's
universals might have been exemplified by its objects very
differently. If they had — if the world's objects and
universals had combined in a very different way — there would
have been a very different set of atomic facts and, hence, a very
different world.

To spell out the idea of a possible fact, the combinatorialist
introduces the more general notion of an atomic (combinatorial)
state of affairs, that is, an entity that simply has the
form of an atomic fact — n objects exemplifying
an n-place relation — but without any requirement that
the exemplification relation in question actually holds between them.
More exactly:

AS

For any objects a1, ...,
an and any n-place relation
R, there is an atomic (combinatorial) state of
affairs a1, ...,
an's exemplifying R (again,
[R,a1,...,an],
for short).

Thus, even if the two hydrogen atoms h1 and
h2 in a water molecule do not in fact stand in the
covalent bonding relation B, there is nonetheless the
(non-factual) state of affairs
[B,h1,h2].

Combinatorialism takes facts to be literal, structured parts of the
physical world. This suggests that a non-factual state of affairs
— a merely possible fact — must be part of a
merely possible physical world. This idea is at odds with the strong,
scientifically-grounded form of actualism that typically motivates
combinatorialism. Two options are available: The combinatorialist can
follow the (actualist) abstractionists and define states of affairs
to be philosophical or mathematical constructs consisting only of
actual objects, properties, relations, and facts. For example, the
state of affairs
[R,a1,...,an] can
simply be identified with the ordered n-tuple
〈R,a1,...,an〉.
So long as the combinatorialist is willing to adopt the additional
metaphysical or set theoretic machinery, this sort of approach offers
a way of introducing non-factual states of affairs that does not
involve any untoward ontological commitments to merely possible
entities. Alternatively, following Armstrong (1989, 46–51;
1997, 172–4), the combinatorialist can refuse to grant
non-factual states of affairs any genuine ontological status and
adopt a form of
modal fictionalism
that nonetheless permits one to speak as if such states of
affairs exist. The exposition to follow will remain largely neutral
between these options.

Constituency for states of affairs is understood as for facts.
Additionally, analogous to molecular facts, there are molecular
states of affairs — conjunctions of atomic states of affairs.
Inclusion between states of affairs is understood exactly as it is
between facts and being a bare particular in a molecular
state of affairs s is understood as for facts: a is a
bare particular in s if there is no monadic conjunct of
s of the form a exemplifies F. The notion of
recombination is now definable straightaway:

s is a recombination of a molecular state of
affairs f =defs is a molecular state
of affairs whose constituent objects and constituent
universals are exactly those of f. s is a
non-trivial recombination of f if it does
not include the same states of affairs as f.

Very roughly then, a possible world will be a certain sort of
recombination of (some portion of) the actual world, the
molecular fact that includes all of the atomic facts. This idea will
be refined in the following sections.

Say that a state of affairs is structural if it is atomic and
its constituent universal is structural or it is molecular and
includes a structural state of affairs; and say that it is
simple otherwise. The difference between structural and simple
universals and states of affairs is particularly significant with
regard to the important concept of supervenience (Armstrong
1989, Ch
8).[45]
Entity or entities S supervene on entity or entities R
if and only if the existence of R necessitates that of
S (ibid., 103). (Necessitation here is, of course,
ultimately to be spelled out in terms of combinatorial possible
worlds.) Non-structural states of affairs supervene directly on their
atomic
conjuncts.[46]
However, things are not in general quite so straightforward for
structural states of affairs. For, although structural states of
affairs are ultimately constituted entirely by simple states of
affairs, unlike non-structural states of affairs, structural states
of affairs typically supervene on more than the totality of their
constituents. For, in many cases, whether or not a structural fact
exists depends not only on the existence of its constituent facts but
also on the absence of certain others (Armstrong 1997,
34ff). For example, as noted in our example above, our water molecule
W comprises two further facts in which two hydrogen atoms
h1 and h2 both stand in the
covalent binding relation with an oxygen atom o. However, if
o were to bind with a further hydrogen atom
h3, then, despite the fact that the constituent
facts of W would still hold, W would not be water;
there would be no such fact as W's being
water.[47]
Rather, W would exist only as a complex part of a hydronium
ion; the new binding [B,o,h3] would,
so to say, “spoil” the instantiation of Water.
Thus, more generally, whether or not a structural state of affairs
S exists in a possible world typically requires something over
and above its constituent states of affairs being “welded
together” in the right sort of way (Armstrong, 1997, 36); it
requires also that there be no relevant “spoilers” for
S.[48]
Armstrong draws directly on the initial passages of the
Tractatus[49]
for the necessary apparatus: a structural state of affairs S
in any possible world w, supervenes, not simply on its
constituent atomic states of affairs but on a certain
higher-order state of affairs Tw,
namely, the state of affairs that the (first-order) atomic states of
affairs of w are all the (first-order) atomic states of
affairs and, hence, that w includes no spoilers for
S. Armstrong (ibid., 35, 134–5, 196–201) calls
Tw the totality state of affairs for the
atomic states of affairs of
w.[50]

The idea of possibility being rooted in arbitrary recombinations of
the actual world, rearrangements of its objects and universals, is
intuitively appealing. Clearly, however, not just any such
recombination can count as a possible world. Some states of affairs
are intuitively impossible — [being an elephant,
e], where e is an individual electron, say — and
some pairs of states of affairs, while individually possible, are not
compossible — the states of affairs [having 1kg
mass, a] and [having 2kg mass, a] for a
given object a, or, for a given mereological sum m of
simples, the states of affairs [being a baboon, m] and
[being a hoolock, m]. But nothing that has been said
rules out the existence of recombinations of the actual world —
rearrangements of its objects and universals — that include
such states of affairs. Obviously, however, such recombinations
cannot be thought to represent genuinely possible worlds. Of course,
like the abstractionist, the combinatorialist could simply stipulate
as part of the definition that all legitimate recombinations must be
genuinely possible states of affairs of a certain sort,
genuinely possible recombinations. But this will not do.
For, like concretism, combinatorialism purports to be a
reductive account of modality, an account of possible worlds
that does not depend ultimately on modal notions (see Armstrong 1989,
33).[51]

Here the distinction between simple and structural states of affairs
together with the combinatorialist's strong notion of supervenience
come to the fore. For, given that structural facts supervene on
simple facts and the actual totality fact T@, the
actual world can be defined more parsimoniously as the molecular fact
that includes all the simple atomic facts and the totality
fact T@. And at the level of simples, there are no
limitations whatever on recombination (Wittgenstein 1921,
2.062-2.063); hence, any recombination of simple objects and
universals is by definition considered possible. Thus Armstrong
(1986a, 579):

The simple individuals, properties, and relations may be
combined in all ways to yield possible [simple]
atomic states of affairs, provided only that the form of
atomic facts is respected. That is the combinatorial idea.

Worlds, in particular, can be defined as special cases of such
recombinations, together with appropriate totality facts. To state
this, we need a condition that ensures the existence of a unique
actual world:

States of affairs s and t are identical iff they
include exactly the same states of affairs.

Given this, we have:

The (combinatorial) actual world =def the fact @ that includes
exactly all the simple atomic facts and the totality state of
affairs T@ for the conjunction of those
facts.

AW3

w is a
(combinatorial) possible world =defw is a recombination of the
simple atomic facts of the actual world conjoined with the totality
fact Tw for that
recombination.[52]

Armstrong's ontological commitments are
notoriously rather slippery but, given AW3, a reasonably
complete notion of existence in a world is forthcoming. First, let us
note that, for Armstrong, the “combinatorial idea” yields
a substantial metaphysical thesis, as well, viz., the ontological
free lunch (1986, 12ff), i.e., the thesis that “[w]hat
supervenes is no addition of being”; that “whatever
supervenes ... is not something ontologically additional to the
subvenient entity or entities.” Hence, for Armstrong, it
appears that simple states of affairs and their constituents
exist most fundamentally and that the existence of more complex
entities is in a certain sense derivative. Thus:

Entity a exists
fundamentally in (combinatorial) possible world w
=def (i) a is a
simple state of affairs that w includes or (ii) a is a
constituent or conjunct of an entity that exists fundamentally in
w.

Given this, existence in a world
generally — both fundamental and derivative — both for
simples and
(first-order[53])
non-simples alike, is definable as follows:

AE3

Entity a exists in (combinatorial) possible world w
=def either (i) a
exists fundamentally in w or (ii) a supervenes on
entities that exist in w.

Semantics
receives rather short shrift in Armstrong's version of
combinatorialism — at least, semantics in the model theoretic
sense of
§1.2
— but, as it has played an important role in our discussion of
concretism and abstractionism, we note briefly how the ontology of
combinatorialism might be taken to populate a possible world
interpretation of the language of modal predicate logic.
Specifically, we can take the range of the modal operators —
understood, semantically, as quantifiers — to be all of the
combinatorial possible worlds in the sense of AW3. The domain
d(w) of each world w is the set of all simple
and complex objects that exist in w according to AE3
and the w-extension Iπ(w) of a
predicate π expressing a simple or complex universal R is
the set of all n-tuples, 〈a1, ...,
an〉 such that the atomic fact
[R,a1,...,an]
exists in w.

There are, then, for the combinatorialist no intrinsically modal
phenomena; there are just all of the various worlds that exist on
unrestricted combinatorial grounds alone. Ultimately, all genuine
possibilities, simple or not, are just states of affairs that exist
in these combinatorial worlds in the sense of AE3. However, it
is not immediately as clear how to understand many intuitive
necessities/impossibilities involving complex
structural universals, for example, the impossibilities noted in the
previous section, viz., that something simultaneously have a mass of
both 1kg and 2kg or simultaneously be both a baboon and a hoolock.
Likewise, it is not entirely clear how combinatorialism accounts for
intuitive facts about essential properties, such as that our water
molecule W is essentially water or that Algol is essentially a
dog. Combinatorialists argue that such modal facts can nevertheless
be explained in terms that require no appeal to primitive modal
features of the world (Armstrong 2004b, 15).

Analytic Modalities. Armstrong argues that many
intuitive modal facts — notably, the impossibility of an object
exemplifying more than one determinate of the same determinable
— can be understood ultimately as logical, or analytic,
modalities that are grounded in meaning rather than any primitive
modal features of reality. For example, intuitively it is impossible
that an object simultaneously exemplify the structural properties
having 2kg mass and having 1kg mass. The combinatorial
reason for this (cf. Armstrong 1989, 79) is that, for an object
a to exemplify the former property is simply for there to be a
division of a into two wholly distinct parts, both of which
exemplify the latter property. Moreover, this division into parts is
entirely arbitrary, that is, for any part a1
of a exemplifying having 1kg mass, there is a (unique)
part a2 of a wholly distinct from
a1 that also exemplifies that property. It follows
that, if our 2kg object a itself also exemplifies
having 1kg mass, then, as a is a part of itself, there
must be a 1kg part of a that is wholly distinct from a.
And that is analytically false, false “solely by virtue of the
meaning we attach to” the word ‘part’
(ibid.,
80).[54]

Emergent Modalities. Combinatorialism purports to
explain a further class of intuitive modal facts as features that
simply “emerge” from facts about structural
properties.[55]
The discussion of structural states of affairs and supervenience
above provides an example. Let us suppose the actual world
w1 includes our water molecule W from Figure
1 plus a further hydrogen atom h3. In this world,
only h1 and h2 bind to o.
Hence, this world includes the state of affairs W's being
water but not the state of affairs I's being hydronium in
which o, h1, h2, and
h3 are so bonded as to constitute a hydronium ion
I. Conversely, however, given the unrestricted nature of
recombination, there is a world w2 that includes
W structured as it actually is in w1 but
which also includes the spoiler
[B,o,h3] — where o and
h3 bond — and, hence, the structural state of
affairs I's being hydronium. Thus, the absence of
[B,o,h3] in w1
enables the emergence of W's being water and precludes I's
being hydronium whilst its presence in w2
enables the emergence of the latter but precludes the former. As a
consequence, it is impossible that the states of affairs W's
being water and I's being hydronium
coexist.[56]

Figure 2: W's being water and (given a bond between o
and h3) I's being hydronium

Although more dramatic, large-scale examples of
incompatible states of affairs — such as a thing's being
simultaneously both a baboon and a hoolock — might be vastly
more complex, there is no obvious reason why their impossibility
could not have the same sort of combinatorial explanation.

Essential Properties. It follows from the
unrestricted nature of recombination that, for any simple object
a and simple universal P, a recombines with
P in some worlds and fails to recombine with P in
others. Generalizing from this fact, it follows that no simple or sum
of simples has any simple universal or conjunction of simple
universals essentially. It also follows that no such object has any
structural property essentially. For assume o is such an
object and that it exemplifies a structural property P. Since
P is structural, it supervenes on some set of simple states of
affairs. But by the nature of recombination, there are combinatorial
worlds in which those states of affairs do not exist and, hence, in
which P doesn't but o — being a simple or a sum
of simples — does.

Thick particulars like our water molecule W don't fare much
better because of the possibility of spoilers. For Armstrong (1997,
35), W is simply the conjunction of its constituent states of
affairs. As we've just seen, however, in the presence of spoilers,
that conjunction would exist — hence, W would exist
— without being Water. Hence, it would seem that at
least some properties that, intuitively, are essential to their
bearers turn out not to be for the combinatorialist. The problem is
compounded by the fact that some intuitively non-essential properties
of some thick particulars are arguably essential for the
combinatorialist. The shape properties of a thick particular
A, for example, would seem to be a function of its constituent
states of affairs. Moreover, the exemplification of such properties
are not obviously subject to spoilers the way that natural kind
properties like Water are. Hence, as A is identical to
the conjunction of its constituent states of affairs, it would seem
that it will have the same shape in any world in which it exists,
i.e., it will have that shape essentially.

That said, combinatorialism can arguably provide a reasonably robust
analysis of intuitions about the essential properties of ordinary
thick particulars like dogs or persons. Such objects can be taken to
be temporal successions of sums of simples and each sum in the
succession as its temporal parts. Sums in the same the rough temporal
neighborhood are composed of roughly the same simples and are
structured in roughly the same way. Similarities between such objects
across worlds in turn determine counterpart relations. Following
Lewis, the essential properties of such objects can then be
identified with those properties exemplified by (all of the temporal
parts of) all of its counterparts in every world in which it exists
(Armstrong 1997, 99–103,
169).[57]

Since a possible world is a recombination of the actual world and
every recombination includes states of affairs involving every simple
individual and every simple universal, by AE3, every simple
entity exists in every world. Hence, there could not have been fewer
of them; nor could there have been simples other than the ones there
actually are. In this section, we address this issue and the issue of
contingent existence generally in combinatorialism.

Fewer things. Combinatorialism as it stands has no
problem accounting for the general intuition that there could have
been fewer things. We have already noted in
§2.3.3
and again in
§2.3.5
how our water molecule W, as such, might not have existed.
More generally, given the unrestricted nature of recombination, for
any a involving a structural fact S, there are
recombinations of the actual world wherein either (a) some of the
relations among a's constituents that are critical to
S's structure fail to be exemplified by those constituents, or
(b) there are further states of affairs included by those
recombinations that act as spoilers for S. Consequently, the
combinatorialist seems to have no difficulty explaining how there
might have been fewer water molecules, humans, etc.

Intuitively, however, there isn't anything in the idea of a simple
that suggests that simples are necessary beings — especially
if, as combinatorialists generally agree, simples are physical things
of some sort and simple universals are properties of, and relations
among, those things. For there is nothing in the nature of a simple
object to suggest that any given simple had to have existed.
Likewise, there is nothing in the nature of a simple universal to
suggest it had to have been exemplified and, hence, on the
combinatorialist's own conception of universals, that it had to
exist. Otherwise put, as simples exist only insofar as they are
constituents of facts, there seems no reason why there couldn't have
been a very small number of facts, indeed, just a single simple,
atomic, monadic fact and, hence, a lone simple object and a lone
simple universal.

In fact, however, AW3 can be easily modified to accomodate
these intuitions without doing any serious violence to
combinatorialist intuitions. Specifically, the combinatorialist can
admit “contracted” worlds in which fewer simples exist by
allowing any recombination of any simple fact — that
is, equivalently, by allowing any state of affairs — to count
as a possible world:

AW3′

w is a (combinatorial) possible world =defw is a recombination of some
simple fact f conjoined with the totality state of affairs
Tw for that recombination.

AE3 requires no modification, as it was defined with
sufficient generality above. Under AW3′, however,
AE3 entails that all entities alike — objects and
universals, simple and structural — are contingent and, indeed,
that every simple object is the sole constituent of some
combinatorial possible world.

Other things. Intuitively, not only could there
have been fewer things, there could have been more things or, more
generally, things other than those that actually exist. As
above, combinatorialism as it stands seems able to account for many
instances of this intuition: Figure 2 illustrates how a non-actual
hydronium ion I might exist in another world. Likewise, there
seems no reason to deny, e.g., that there are rearrangements w
of the actual world's simples wherein exist all of the human beings
that actually exist (at, say, 0000GMT 1 January 2013) and more
besides that are composed of simples that, in fact, constitute things
other than human beings (Armstrong 1997,
165).[58]
Combinatorialism also seems able to account for the possibility of
conjunctive and structural universals that are simply rearrangements
of actual simples. It is not implausible to think that such
recombinations can give rise to, say, exotic biological kinds that
have no actual instances (Armstrong 1989, 55–56). Thus, in
particular, combinatorialism seems quite able to provide the truth
condition
(24)
for
(23)
and, hence, can account for some possibilities involving
“missing” universals that, intuitively, ought to be
possible.

However, it is far from clear that such possibilities exhaust the
modal intuition that other things could have existed. Notably,
intuitively, there could have been different simple
universals distinct from any that actually exist — different
fundamental properties of simples, for example. Likewise for simple
objects. Either way, there seems to be nothing in the idea of a
simple object or simple universal that suggests there couldn't have
been simples other than, or in addition to, the simples there are in
fact. But AW3′ does not allow for this; the simples of
every possible world are a subset of the actual simples and there is
no obvious way of modifying the principle to accommodate the
intuition. Nor is there any obvious way of modifying the principle to
accomodate the intuition in
question.[59]

The combinatorialist could of course abandon actualism and include
merely possible simples into her ontology. Again, she could follow
the new actualists and draw a division between actually concrete and
non-actual, possibly concrete simples; or she could introduce
Plantinga-style haecceities to go proxy for merely possible simples.
But all of these options would be badly out of step with the strong,
naturalist motivations for combinatorialism: There is but the one
physical world comprising all of the facts; recombinations of (at
least some of) of those facts — arbitrary rearrangements of
their simple objects and universals — determine the possible
worlds. Mere possibilia, merely possible non-concretia,
and non-qualitative haecceities have no real place in that picture.

The “purest” option for the combinatorialist is simply to
brazen it out and argue that the actual simples are, in fact, all the
simples there could be (Armstrong 1989, 54ff; Driggers 2011, 56–61).
A more robust option suggested by Skyrms (1981) makes some headway
against the problem by introducing an “outer”, or
“second-grade” realm of possibility, but at the cost of
moving beyond the basic intuitions of combinatorialism (Armstrong
1989, 60; 1997, 165–167). Finally, Sider (2005, 681) suggests that
combinatorialists who (like Armstrong) are modal fictionalists can
deal with the problem of missing entities simply by appealing to yet
more fictionalism: As the combinatorialist fiction already
includes non-actual states of affairs with actually existing
constituents, there seems no reason not to extend the fiction to
include non-actual states of affairs whose constituents include
non-actual particulars and universals. Fictionalism itself, however,
leaves the combinatorialist with the deep problems detailed by Kim
(1986), Lycan (1993), and Rosen
(1993).[60]

As with concretism and abstractionism, combinatorialism provides
reasonably clear definitions of possible worlds and existence in a
world and is noteworthy for its attempt to avoid what might be
thought of as the metaphysical excesses of the two competing views.
In contrast to concretism, combinatorialism is staunchly actualist:
instead of an infinity of alternative physical universes, each with
its own unique inhabitants existing as robustly as the inhabitants of
the actual world, the worlds of combinatorialism are simply
rearrangements of the universals and particulars of the actual world;
and commitment even to them might be avoided if some version
fictionalism is tenable. Likewise, in contrast to abstractionism's
rather rich and unrestrained ontology of SOAs, combinatorialism's
states of affairs are comparatively modest. Moreover, in contrast to
nearly all versions of abstractionism, combinatorialism shares with
concretism the virtue of a reductive theory of modality: Modal
statements, ultimately, are true or false in virtue of how things
stand with respect to worlds that are themselves defined in non-modal
terms.

Combinatorialism's ontological modesty, however, is also a weakness.
For, unlike, the two competing approaches, there are modal intuitions
that the combinatorialist is not easily able to account for, notably,
the intuition that there could have been other things. Additional
difficulties are discussed in the supplemental document
Further Problems for Combinatorialism.

Goldblatt, R., 2003. “Mathematical Modal Logic: A View of
its Evolution”, in D. M. Gabbay and J. Woods (eds.),
Handbook of the History of Logic, Vol. 7: Logic and the Modalities
in the Twentieth Century. Amsterdam: Elsevier, pp.
1–98.

Nortmann, U., 2002. ‘The Logic of Necessity in Aristotle: An
Outline of Approaches to the Modal Syllogistic, Together with a
General Account of de dicto- and de
re-Necessity’, History and Philosophy of Logic, 23:
253–265.

Wittgenstein, L., 1921. ‘Logisch-Philosophische
Abhandlung’, with a forward by Bertrand Russell, Annalen der
Naturphilosophie, 14, published by Wilhelm Ostwald, Leipzig:
Verlag Unesma: 185–262. Also available online in HTML, PDF,
and ePub formats in a
side-by-side presentation with the translations Wittgenstein (1922)
and Wittgenstein (1974) at
http://people.umass.edu/klement/tlp/.

Acknowledgments

The author wishes to express his deep gratitude to Phillip Bricker and
Max Cresswell for extensive comments on several drafts of this entry
and for numerous illuminating discussions of its content and related
topics. The entry is vastly better for their generous input. Errors
and other infelicities that remain are of course the sole
responsibility of the author. A great deal of this entry was written
with the support of the Alexander von Humboldt Foundation while the
author was a Visiting Fellow at the Munich Center for Mathematical
Philosophy in 2011–12. Thanks are due to the Center's director,
Professor Hannes Leitgeb, for making the author's stay at this
remarkable venue possible. Finally, the author would like to express
his thanks to the SEP Editors for their extraordinary patience in
dealing with the very tardy author of a badly-needed entry.